Properties

Label 8280.2.a.bp
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1161328736\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + (\beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + (\beta_1 + 1) q^{7} + (\beta_{2} + \beta_1 + 2) q^{11} + (\beta_{2} + \beta_1) q^{13} + (\beta_1 + 1) q^{17} + (\beta_{2} - \beta_1 + 2) q^{19} - q^{23} + q^{25} + (\beta_{2} - 1) q^{29} + ( - \beta_{2} - 1) q^{31} + (\beta_1 + 1) q^{35} + ( - \beta_1 + 1) q^{37} + (\beta_{2} + 5) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{2} - \beta_1 + 6) q^{47} + (\beta_{2} + 4 \beta_1 - 2) q^{49} + (\beta_1 + 1) q^{53} + (\beta_{2} + \beta_1 + 2) q^{55} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{59} + ( - 3 \beta_{2} - \beta_1 + 4) q^{61} + (\beta_{2} + \beta_1) q^{65} + ( - 4 \beta_{2} - \beta_1 + 1) q^{67} + ( - 3 \beta_{2} + 4 \beta_1 + 1) q^{71} + ( - 3 \beta_{2} - 5 \beta_1 + 4) q^{73} + (\beta_{2} + 5 \beta_1 + 4) q^{77} + 4 \beta_{2} q^{79} + ( - 2 \beta_{2} - \beta_1 + 3) q^{83} + (\beta_1 + 1) q^{85} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{89} + (\beta_{2} + 3 \beta_1 + 2) q^{91} + (\beta_{2} - \beta_1 + 2) q^{95} + (2 \beta_{2} - 2 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} + 4 q^{7} + 6 q^{11} + 4 q^{17} + 4 q^{19} - 3 q^{23} + 3 q^{25} - 4 q^{29} - 2 q^{31} + 4 q^{35} + 2 q^{37} + 14 q^{41} + 16 q^{47} - 3 q^{49} + 4 q^{53} + 6 q^{55} + 4 q^{59} + 14 q^{61} + 6 q^{67} + 10 q^{71} + 10 q^{73} + 16 q^{77} - 4 q^{79} + 10 q^{83} + 4 q^{85} - 20 q^{89} + 8 q^{91} + 4 q^{95} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.76156
−0.363328
3.12489
0 0 0 1.00000 0 −0.761557 0 0 0
1.2 0 0 0 1.00000 0 0.636672 0 0 0
1.3 0 0 0 1.00000 0 4.12489 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8280.2.a.bp yes 3
3.b odd 2 1 8280.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8280.2.a.bk 3 3.b odd 2 1
8280.2.a.bp yes 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{3} - 4T_{7}^{2} - T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{3} - 6T_{11}^{2} + 2T_{11} + 20 \) Copy content Toggle raw display
\( T_{13}^{3} - 10T_{13} + 8 \) Copy content Toggle raw display
\( T_{17}^{3} - 4T_{17}^{2} - T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4T^{2} - T + 2 \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} + 2 T + 20 \) Copy content Toggle raw display
$13$ \( T^{3} - 10T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 4T^{2} - T + 2 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 14 T - 8 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 3 T - 10 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 7 T - 4 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} - 5 T + 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} + 57 T - 64 \) Copy content Toggle raw display
$43$ \( T^{3} - 40T + 64 \) Copy content Toggle raw display
$47$ \( T^{3} - 16 T^{2} + 66 T - 80 \) Copy content Toggle raw display
$53$ \( T^{3} - 4T^{2} - T + 2 \) Copy content Toggle raw display
$59$ \( T^{3} - 4 T^{2} - 67 T - 142 \) Copy content Toggle raw display
$61$ \( T^{3} - 14 T^{2} - 2 T + 68 \) Copy content Toggle raw display
$67$ \( T^{3} - 6 T^{2} - 109 T - 20 \) Copy content Toggle raw display
$71$ \( T^{3} - 10 T^{2} - 199 T + 1868 \) Copy content Toggle raw display
$73$ \( T^{3} - 10 T^{2} - 130 T + 764 \) Copy content Toggle raw display
$79$ \( T^{3} + 4 T^{2} - 128 T - 256 \) Copy content Toggle raw display
$83$ \( T^{3} - 10 T^{2} + 3 T + 4 \) Copy content Toggle raw display
$89$ \( T^{3} + 20 T^{2} + 56 T + 32 \) Copy content Toggle raw display
$97$ \( T^{3} + 10 T^{2} - 44 T - 472 \) Copy content Toggle raw display
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